An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB®

247 Pages·2011·3.27 MB·English

by  Sebastian Aniţa

ModelingandSimulationinScience,EngineeringandTechnology
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Sebastian Ani¸ta
Viorel Arna˘utu
Vincenzo Capasso
An Introduction to Optimal
Control Problems in Life
Sciences and Economics
From Mathematical Models to Numerical
Simulation with MATLAB(cid:2)R
SebastianAni¸ta VincenzoCapasso
FacultyofMathematics ADAMSS(InterdisciplinaryCentre
University“Al.I.Cuza”Ias¸i forAdvancedAppliedMathematical
Bd.CarolI,11 andStatisticalSciences)
and and
InstituteofMathematics DepartmentofMathematics
“OctavMayer”Ias¸i Universita`degliStudidiMilano
Bd.CarolI,8 ViaSaldini50
Romania 20133Milano
[email protected] Italy
[email protected]
ViorelArna˘utu
FacultyofMathematics
University“Al.I.Cuza”Ias¸i
Bd.CarolI,11
700506Ias¸i,Romania
[email protected]
ISBN978-0-8176-8097-8 e-ISBN978-0-8176-8098-5
DOI10.1007/978-0-8176-8098-5
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2010937643
MathematicsSubjectClassification(2010):49-XX,49J15,49J20,49K15,49K20,49N25,65-XX,65K10,
65L05,65L06,68-04,91Bxx,92-XX
(cid:2)c SpringerScience+BusinessMedia,LLC2011
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To our families

Preface
ControltheoryhasdevelopedrapidlysincethefirstpapersbyPontryaginand
collaboratorsinthelate1950s,andisnowestablishedasanimportantareaof
applied mathematics. Optimal control and optimization theory have already
found their way into many areas of modeling and control in engineering, and
nowadays are strongly utilized in many other fields of applied sciences, in
particular biology,medicine, economics,and finance. Researchactivity in op-
timal controlis seen as a source of many useful and flexible tools, such as for
optimal therapies (in medicine) and strategies (in economics). The methods
of optimal control theory are drawn from a varied spectrum of mathematical
results,and,ontheotherhand,controlproblemsprovidearichsourceofdeep
mathematical problems. The choice of applications to either life sciences or
economicstakesintoaccountmoderntrendsoftreatingeconomicproblemsin
osmosis with biological paradigms.
A balance of theory and applications, the text features concrete exam-
ples of modeling real-world problems from biology, medicine, and economics,
illustrating the power of control theory in these fields.
The aim of this book is to provide a guided tour of methods in optimal
control and related computational methods for ODE and PDE models, fol-
lowing the entire pathway from mathematical models of real systems up to
computer programs for numerical simulation. There is no pretense of being
complete; the authors have chosen to avoidas much as possible technicalities
that may hide the conceptual structure of the selected applications. A fur-
ther important feature of the book is in the approachof “learning by doing.”
The primary intention of this book has been to familiarize the reader with
basic results and methods of optimal control theory (Pontryagin’s maximum
principle and the gradient methods); it provides an elementary presentation
ofadvancedconceptsfromthe mathematicaltheoryofoptimalcontrol,which
are necessary in order to tackle significant and realistic problems. Proofs are
produced whenever they may serve as a guide to the introduction of new
conceptsandmethodsintheframeworkofspecificapplications,otherwiseex-
plicit references to the existing literature are provided. “Working examples”
VII
VIII Preface
are conceived to help the reader bridge those introductory examples fully
developed in the text to topics of current research. They may stimulate
Master’s and even PhD theses projects.
The computer programs are developed and presented in MATLAB(cid:2)R
which is a product of The MathWorks, Inc. This is a very flexible and simple
programming tool for beginners, but it can also be used as a high-level one.
The numerical routines and the GUI (Graphical Users Interface) are quite
helpful for programming. Starting with simple programs for simple models
we progress to difficult programs for complicated models. The construction
of every program is carefully presented. The numerical algorithms presented
here have a solid mathematical basis. One of the main goals is to lead the
readerfrommathematicalresultstosubsequentMATLAB programsandcor-
responding numerical tests.
The volume is intended mainly as a textbook for Master’s and graduate
courses in the areas of mathematics, physics, engineering, computer science,
biology, biotechnology, and economics. It can also aid active scientists in the
above areas whenever they need to deal with optimal control problems and
related computational methods for ODE and PDE models.
Chapter 1 is devoted to learning several MATLAB features by examples.
AsimplemodelfromeconomicsispresentedinSection1.1.1,andmodelsfrom
biology may be found in Sections 1.5 and 1.7. Chapter 2 deals with optimal
controlproblemsgovernedbyordinarydifferentialequations.ByPontryagin’s
principlemoreinformationaboutthestructureofoptimalcontrolisobtained.
Computer programs based on mathematical results are presented. Chapter 3
isdevotedtonumericalapproximationbythegradientmethod.Herewelearn
to calculate the gradient of the cost functional and to write a corresponding
program.Chapter4concernsage-structuredpopulationdynamicsandrelated
optimalharvestingproblems.Chapter5dealswithsomeoptimalcontrolprob-
lemsgovernedbypartialdifferentialequationsofreaction–diffusiontype.The
last two chapters connect theory with scientific research.
Basic concepts and results from functional analysis and ordinary differen-
tial equations including Runge–Kutta methods are provided in appendices.
Matlabcodes,ErrataandAddendacanbefoundatthepublisher’swebsite:
http://www.birkhauser-science.com/978-0-8176-8097-8.
We wish to thank Professor Nicola Bellomo, Editor of the Modeling and
Simulation in Science, Engineering, and Technology Series, and Tom Grasso
from Birkha¨user for supporting the project.
Last but not the least, we cannot forget to thank Laura-Iulia [SA], Maria
[VA], and Rossana [VC], for their patience and great tolerance during the
preparation of this book.
Ia¸si and Milan Sebastian Ani¸ta
May, 2010 Viorel Arna˘utu
Vincenzo Capasso
Symbols and Notations
IN the set of all nonnegative integers
∗
IN the set of all positive integers
Z the set of all integers
IR the real line (−∞,+∞)
IR∗ IR\{0}
IR+ or IR+ the half-line [0,+∞)
IR∗ the interval (0,+∞)
+
n
IR the n-dimensional Euclidean space
x·y the dot product of vectors x,y ∈IRn
(cid:5)·(cid:5)X the norm of a linear normed space X
∂h
∇h, hx, the gradient of the function h
∂x
∂h
hx, the matrix of partial derivatives of h with respect
∂x
to x=(x1,x2,...,xk)
A∗ the adjoint of the linear operator A
Ω ⊂IRn an open subset of IRn
Lp(Ω),1≤p<+∞ the space of all p-summable functions on Ω
L∞(Ω) the space of all essentially bounded functions on Ω
Lp(0,T;X) (X a Banach space) the space of all p-summable
functions (if 1≤p<+∞), or of all essentially
bounded functions (if p=+∞), from (0,T) to X
p
L (0,T;X) (X a Banach space) the space of all locally
loc
p-summable functions (if 1≤p<+∞), or of all
locally essentially bounded functions (if p=+∞),
from (0,T) to X
L∞([0,A)) the set of all functions from [0,A) to IR
loc
belonging to L∞(0,A˜), for any A˜∈(0,A)
L∞([0,A)×[0,T]) the set of all functions from [0,A)×[0,T] to IR
loc
belonging to L∞((0,A˜)×(0,T)), for any
A˜∈(0,A)